ON GENERATING LOOPS OF THE LINK OF A WAHL SINGULARITY

Abstract

Let us consider the first homology group H (L, Z) of the link of a Wahl singularity. According to Mumford, H (L, Z) is generated by suitably oriented loops around the exceptional curves in the minimal resolution. It can be easily seen that the loop around one of the end branches of the exceptional curves generates H (L, Z). However, it is not always true that the loop around an intermediate curve generates H (L, Z). In this article, we define a special intermediate curve (which we call an ancestor ) in the chain of exceptional curves, and prove the the loop around the ancestor generates H (L, Z). This can be applied to the construction of exceptional vector bundles on Dolgachev surfaces which are constructed via Q-Gorenstein smoothing.